摘 要：数值逼近作为函数拟合中的核心工具，在科学计算与工程应用中具有重要意义，其方法选择直接影响拟合精度与效率。本研究以多项式插值、最小二乘法及样条逼近为对象，系统比较了三种方法在不同数据分布和噪声条件下的适用性与性能表现。通过构造典型测试函数并引入高斯噪声模拟实际数据误差，采用均方误差、拟合偏差及计算复杂度等多指标评估体系，定量分析各方法的优劣。研究表明，多项式插值在光滑数据下表现优异，但对高次多项式存在龙格现象；最小二乘法对含噪声数据具有较强鲁棒性，适合大规模数据拟合；样条逼近则在局部变化复杂的函数中展现出更高的灵活性与精确性。本研究创新性地结合理论分析与实验验证，提出基于数据特性的方法选择策略，为实际问题中的函数拟合提供了重要参考依据，同时为进一步优化数值逼近算法奠定了基础。
关键词：数值逼近；最小二乘法；样条逼近；多项式插值；方法选择策略
A Comparative Study of Numerical Approximation Methods in Function Fitting
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Abstract：Numerical approximation, as a core tool in function fitting, plays a significant role in scientific computing and engineering applications, where the choice of method directly affects the accuracy and efficiency of fitting. This study systematically compares polynomial interpolation, least squares, and spline approximation regarding their applicability and performance under different data distributions and noise conditions. By constructing typical test functions and introducing Gaussian noise to simulate real-world data errors, a multi-criterion evaluation system incorporating mean square error, fitting deviation, and computational complexity is employed to quantitatively analyze the advantages and disadvantages of each method. The results indicate that polynomial interpolation performs excellently with smooth data but suffers from Runge's phenomenon for high-degree polynomials; the least squares method demonstrates strong robustness for noisy data and is suitable for large-scale data fitting; spline approximation exhibits superior flexibility and precision in functions with complex local variations. Innovatively combining theoretical analysis with experimental validation, this study proposes a strategy for method selection based on data characteristics, providing crucial reference for function fitting in practical problems and laying a foundation for further optimization of numerical approximation algorithms.
Keywords: Numerical Approximation；Least Squares Method；Spline Approximation；Polynomial Interpolation；Method Selection Strategy
目 录
引言 1
一、数值逼近方法概述 1
(一)常见数值逼近方法介绍 1
(二)数值逼近的基本原理 2
(三)方法在函数拟合中的意义 2
二、插值法在函数拟合中的应用 3
(一)多项式插值的理论基础 3
(二)分段插值的适用性分析 3
(三)插值法的误差与优化策略 4
三、最小二乘法的应用比较 4
(一)最小二乘法的基本原理 4
(二)线性与非线性最小二乘法对比 5
(三)最小二乘法在复杂数据中的表现 5
四、样条函数与径向基函数的比较 6
(一)样条函数的构造与特性 6
(二)径向基函数的拟合能力分析 6
(三)两种方法的优劣与适用场景 7
结论 7
参考文献 9
致谢 9
